Question

Given z equals quantity negative 11 minus 4 times I end quantity over quantity 3 plus 5 times i end quantity comma rewrite z in standard form.

Answer 1

To rewrite z = (-11 - 4i) / (3 + 5i) in standard form, multiply the numerator and denominator by the conjugate of the denominator to remove the imaginary unit from the denominator. The result is z = -53/34 + (43/34)i, which is the complex number in standard form.

Step-by-step explanation:

A student requested assistance to rewrite the complex number z given by the expression z = (-11 - 4i) / (3 + 5i) into standard form. The standard form of a complex number is a + bi, where a and b are real numbers, and i is the imaginary unit (i^2 = -1).

To convert the complex fraction into standard form, we must first eliminate the imaginary number from the denominator. We do this by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi. Thus, the conjugate of the denominator 3 + 5i is 3 - 5i.

The multiplication of the fraction by its conjugate yields:

• Multiply the numerator: (-11 - 4i)(3 - 5i) = -33 + 55i - 12i + 20i^2.
• Since i^2 = -1, we simplify the expression to -33 + 43i - 20 = -53 + 43i.
• Multiply the denominator: (3 + 5i)(3 - 5i) = 9 - 15i + 15i - 25i^2.
• Since i^2 = -1, the denominator simplifies to 9 + 25 = 34.

Therefore, the fraction is:

z = (-53 + 43i) / 34

To divide each term of the numerator by the denominator, we get:

• Real part: -53 / 34
• Imaginary part: 43i / 34

So in standard form, z is:

z = -53/34 + (43/34)i

This is the complex number in standard form, but it may be further simplified by division if necessary.